A decision problem $P$ over $L$ is defined by a relation $R$ between two witness counts under a side condition $S$ when there are two existential second-order sentences over $L \cup \{\le\}$ whose witness counts $c$ and $d$ satisfy, on every nonempty finite $L$-structure with every linear order, $S(c, d)$, and $P$ holds exactly when $R(c, d)$. The class of $S$ and $R$ consists of these problems, with cofinal hardness. Taking for the answer a property of the counts, rather than of one integer, is the two-number form of the GapP characterization of Fenner, Fortnow, and Kurtz.

The classes are ⊕P, after Papadimitriou and Zachos and, independently, Goldschlager and Parberry, where the first count is odd; Mod$_k$P, after Cai and Hemachandra, where it is not a multiple of $k$; PP, after Gill, where the first count exceeds the second; C$_=$P, after Wagner, where the two counts are equal; and UP, after Valiant, where the first count is at most one and the answer is whether it is one. No complete problem is known for UP, and the question does not relativize, as Hartmanis and Hemachandra showed: UP is here for its inclusions. The decision version of a counting problem by a property of numbers holds on the structures whose count has the property.